Metric response and Hermitian Yang–Mills flow for two-step extensions

Abstract

For two-step extensions of pairwise nonisomorphic stable bundles on a compact Kähler manifold, we quantify the degeneration of Hermitian–Einstein metrics and Hermitian Yang–Mills flow near intersecting slope-stability walls. We assume equal slopes at the reference form and connected support for the extension class. When the centered slopes provide full local parameters, every singular value of the trace-free part of $H^{-1}dH$, for the Hermitian–Einstein metric $H$, is relatively asymptotic to the reciprocal of a nonzero eigenvalue of a weighted extension-graph Laplacian. The comparison uses rank-weighted slope and normalized $L^2$ norms and remains uniform for arbitrarily separated positive stability margins. We also construct a finite-dimensional invariant manifold of the flow and compare its solutions with flows from any smooth family of initial metrics. For each tolerance, one late time and then one parameter neighborhood give multiplicative metric comparison, up to a constant scalar, and relative accuracy for every small endomorphism Dolbeault eigenvalue for all subsequent times. The comparison applies in the stable, semistable and unstable regimes. In the two-factor case, we determine a uniform asymptotic profile for the smallest eigenvalue across the wall, including exponential decay on the unstable side. An application to explicit weak Fano toric tangent bundles connects nonsimplicial stability chambers, metric response and flow on the same bundles.